Squeezing functions and Cantor sets

Abstract

We construct large Cantor sets whose complements resemble the unit disk arbitrarily well from the point of view of the squeezing function, and we construct large Cantor sets whose complements do not resemble the unit disk from the point of view of the squeezing function. Finally we show that complements of Cantor sets arising as Julia sets of quadratic polynomials have degenerate squeezing functions, despite of having Hausdorff dimension arbitrarily close to two.